Synthesis of Driving-Point Impedances with Active RC Networks By I. W. SANDBERG (Manuscript received December 4, 1959) A general method is presented for synthesizing driving-point impedances using RC networks and active elements. The procedure realizes any real rational driving-point function and leads to rather simple structures. Only one active device, a negative-impedance converter, is required. The synthesis of biquadratic impedance functions is considered in detail, I. INTRODUCTION It is often desirable to avoid the use of magnetic elements in synthesis procedures, since resistors and capacitors are more nearly ideal elements and are usually cheaper, lighter and smaller. This is especially true in control systems in which, typically, exacting performance is required at very low frequencies. The rapid development of the transistor has pro- vided the network synthesist with an efficient low-cost active element and has stimulated considerable interest in active RC network theory during the past decade. '" ' The present paper considers the active RC synthesis of driving-pomt impedances. Transfer functions are not treated directly, but are covered at least in principle, since it is always possible, and indeed sometimes convenient, to reduce the synthesis of transfer functions to the synthesis of two-terminal impedances. It is now well known that any driving-point impedance function ex- pressible as a real rational fraction in the complex frequency variable can by synthesized as an active RC network requiring only one ideal active element. Two proofs of this result are already in the literature."'^* The present paper provides a third proof, although its main objective is to present a new and more practical realization network. The synthesis * Anotlier proposed proof* ia in fact concerned only with thoae impediince functions which are positive on some section of the negative- real axis of the complex frequency plane. 947 948 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1960 technique is similar to that used in the author's first proof,^ and con- siderably simpler than those used in the later paper by Kinariwala.' Like the technique previously presented by the author,^* the present technique yields the values of the required networlc elements explicitly and directly, without the solution of simultaneous equations. The gen- eral procedure yields networks that contain more capacitors than the absolute minimum required; in return for this, however, it yields struc- tures which, unlike those of Kinariwala, do not need balanced amplifiers or complex resistance networks, f An alternative procedure applicable to a wide class of biquadratic impedance functions is also described. This procedure leads to structures requiring only two capacitors. The synthesis techniques presented in this paper are based on net- works employing a type of impedance converter that is a generalization of the negative-impedance converter. The converter concept is intro- duced in a general way in order to properly orient the reader. II. THE IDEAL IMPEDANCE CONVERTER Consider a two-port network terminated by an impedance Zt(s) as shown in Fig. 1. The input impedance at port 1 is ZM = hn - hnhiiZris) 1 + /l22^r(s) ' (1) where the hybrid parameters are functions of the complex frequency variable defined by -El = huh + huEi , (2) II Iz + + E, Es Zt{s) Fig. 1 — Two-port network. * The particular configuration involved is essentially the one mentioned in Section 3.2 of this paper. t A more practical procedure in Ref. 7 treats a restricted class of driving-point functions and employs a passive RC two-port network terminated by a negative RC impedance. Other restricted realization techniques are presented in Ref. 8. SYNTHESIS OF DRIVING-POINT IMPEDANCES 949 For the ideal impedance converter we require that for every Zt{s) Zi(s) - K{s)Zr{s), (3) where K{s) is a predetermined fixed function of the complex frequency variable. In order that (1) and (3) be compatible, (4) K{s) = -hi2h2i . An ideal impedance converter, therefore, is a two-port network with a hybrid parameter matrix of the form ' /H2 ,/t21 It follows that the impedance at port 2 with the termination Zt{s) connected to port 1 is Zt{s) (5) ZM = K{s)' (6) A "controlled-source" representation of the unbalanced ideal impedance converter is given in Fig. 2. The negative-impedance eonveter (huhoi ^ 1) has been heavily ex- ploited as a synthesis tool m active RC network theory. The synthesis techniques presented in this paper are based on networks employing a more general type of converter characterized by (7) where Z3/Z4 is the ratio of two RC driving-point unpedances. This hy- brid parameter matrix can be realized with either of the ideahzed cir- "0 1 z, z. -32 -f- (h,2-0E; (ha, + Oil I'o 02' Fig. 2 — Representation of the ideal impedance converter. 950 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1960 -o2 1 I Fig. 3 — Idealized realization of the required converter. Fig. 4 — Alternate idealized realization of the converter. cuits shown in Figs. 3 and 4.* The smaller rectangles enclose "hifinite gain" current amplifiers. The realization of Fig. 3 is a modification of Larky's idealized current-inversion negative-impedance converter. The larger rectangle in Fig. 4 encloses Linvill's well-known idealized voltage- inversion negative-impedance converter.^ Hence, (7) can be realized with Linvill's negative-impedance converter and two RC driving-point impedances. A controUed-source representation of (7) is given in Fig. 5. Transistor realizations are discussed in the Appendix. 0(-a'' -3 2' Fig. 5 — Representation of (7). Many other realizations are possible. SYNTHESIS OF DRIVING-POINT IMPEDANCES 951 Fig. G — Realization one-port. III. SYNTHESIS OF DRIVING-POINT IMPEDANCES The driving-point impedance of the one-port network shown in Fig, 6 is* Z(s) = Zi — Za 2i z,- Z2 (8) The impedance function that is to be synthesized is a real rational fraction in the complex frequency variable Z(s) = P(s) (9) The synthesis consists of identifying each of the four parameters Zi , Z2 , Z3 and Zi with a two-terminal RC impedance function. The presence of negative signs in (8) suggests an approach similar to techniques previously proposed for the synthesis of transfer functions.''^ Assume that the prescribed function Z(s) is positive on at least one section of the negative-real axis of the complex frequency plane. Sup- pose we write P(s) Z(s) = n (s + <y.) Qis) (10) n(s + 'T.) where * This expression can be readily obtained by applying Blackman's equation" to the network that results when the Ponverter is replaced by its controlled source representation. The required return differences are computed with respect to the quantity 1 + {Zs/Zt). A detailed study of Blackman's equation led to the dis- covery of the realization networks presented in this paper. 952 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1960 ^ tri < 0-2 < 0-3 ■ ■ ■ < (Tn , and the number A'' is equal to the degree of the polynomial P{s) or of the polynomial Q(s), whichever is greater. The points — o-, are chosen to lie anywhere on the section or sections of the negative-real axis where the function Z{s) is finite and positive.* We replace both the numerator and denominator of the right-hand side of (10) by their partial fraction expansions and group the resulting terms to obtain Z{s) = ^ ^^ (U) Pi P2 Qi Q2 Ps Pa Qi Q2 where N QiQi = n (5 + cTi) and Pi/Qi , P2/Q2 , PJQi , and P4/Q2 are each RC driving-point imped- ances. The function Z{s) is expressible in this form since: (a) Each of the two partial fraction expansions possesses real residues of like sign at any particular pole. (b) All RC driving-point impedances are expressible in the form Rca ~\~ Aj ttk k=i s -\- bk' where R^ , ak and 6t are nonnegative. Equation (11) is equivalent to Pi P ^^'^ - P3 + W1 pI-Trq. - ^^^^ We identify the impedance parameters of (8) as follows: 7 _ Pi 7 _ P2 '^' '^' (13) 7 ^ Pi 7 ^ ^2 ^ P^^RQ^ ' Pa^RQi' * The choice of the ff.'s influences the spread of element values and the sensi- tivity of the driving-point impedance to variations in the active and passive elements. SYNTHESIS OF DRIVIXG-POINT IMPEDANCES 953 According to the qualifications appended to (11), the functions Z3 and Z^ are RC driving-point impedances. Consider the function Zi . The degree of the polynomial Qi is at least as great as the degree of P3 . Note that, as the parameter R assumes nonnegative real values that increase from zero to infinity, the poles of Zi move from the zeros of P3 to the zeros of Qi . Recall that the ratio of polynomials Pi/Qi is an RC driving-point impedance. Consequently, it is always possible to find a finite value of the parameter R for which the poles and zeros of Zi interlace properly. A similar argument shows that Z2 also can always be made an RC driving-point impedance. Note that Z(s) need not be a positive-real function. 3.1 -471 Example Let Z(s)=i±^. (14) This function is positive on the entire negative-real axis. We choose 0-1 = 1, and (72 = 2. From (12), s + 2 _ 3 7/ ■, s + 1 s + 2 /,.N ^^ s(l + fl) + 3 + /2 Rs+±±2R- ' s + 1 s+ 2 Employing (13), ^4 - 1 + ^-T , Z,= '^ s + 1 ' ' s + 2' (16) s(l + /?) + 3 + P' - /2s + 4 + 2/2' The sot of unpedances is realizable for R ^ \. Choose 72 ^ 1 so that The corresponding network is shown in Fig. 7. Note that Zz and Z4 can be scaled by the same constant without af- fecting the impedance Z{s). In a practical design, this degree of freedom would be utilized in order to optimize some figure of merit such as the sensitivity function. The choice of the tr.'s would be similarly influenced. 954 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 19G0 Fig. 7 — Realization of (14). 3.2 An Alternate Realization The class of functions treated in this section can also be synthesized with the network given in Fig. 8. The input impedance of this structure Z(.s) = (18) By employing arguments similar to those already discussed, it can be shown that the value of a nonnegative real parameter R can be chosen to ensure that the following set of impedances is realizable:* Zs = Z, - P, + RQi Pi P3 + fiQl' Z, = z, = p, + RQ2 Q2 ' Pi Pa + RQ2' (19) Fig. 8 — Alternate realization one-port. * A realizable set of impedaacea can also be obtained in other ways. SYNTHESIS OF DRIVING-POINT IMPEDANCES 955 This circuit has the disadvantage of requiring a "floating" two-port active network. 3.3 Reslnclions on Z{s) The methods of synthesis presented in the previous sections are not appHcable to functions that are nonpositive on the entire negative-real axis. This is a significant theoretical restriction, since positive-real func- tions, for example, need not possess the requu-ed property. In particular, all reactance functions must be excluded. The difficulty mentioned above can be circumvented m several ways by modifications of the synthesis technique. Suppose that the prescribed impedance Z{s) is nonpositive on the entire negative-real axis. The function Z'{s) = Z(s) - — ^ ao > 0, b, ^ (20) s -\- bo must, however, be positive on one section of the negative-real axis. It can therefore be synthesized by the previously discussed procedure. The impedance Z(s) is obtained by connecting an RC impedance a^/is 4- bo) in series with the resulting network. An alternative procedure on an admittance basis also applies, the network being modified at the input terminals by the parallel connection of an RC impedance Co + - do>0, Co ^ 0. s Both methods usually necessitate a larger number of passive com- ponents than would be required for the synthesis of —Z(s). For this reason it may be more desirable to employ a negative-impedanc^e con- verter terminated by —Z(s). 3.4 Sufficiency of One Active Element or One Negative-Impedance Converter Since the realization of the converter requires only one active element or only one Linviil-type negative-impedance converter, the preceding discussion constitutes a proof of Theorem: Ani/ driving-point impedance function, expressible as a real rational fraction in the complex frequency variable, can be synthesized with a network containing only resistors, capacitors and either a single ideal ac- tive element or a single ideal negative-impedance converter. Note that it is theoretically possible to synthesize impedance func- tions which approach infuiity as any integral power of the frequency 956 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1960 variable. Obviously, these functions can be realized by actual networks only over the frequency band where the active two-port is essentially characterized by the controlled-source model of Fig. 5.* IV. BIQUADRATIC SYNTHESIS The synthesis of biquadratic impedance functions merits special at- tention. The network associated with a function of this type, previously considered as an example, requires a total of four capacitors (two in the converter). Two of the capacitors can be eliminated by employing an alternative technique based on the network of Fig. 6 with Zi and Zi re- placed by resistors R\ and Ri . The structure becomes an impedance converter imbedded in a simple resistance network. From (8), Zs _ RiJZ — Ri) .^ . Z4 Ri(Z — R2) The biquadratic function Z(s) is given by ^ ( .-.0(s-..) (s~ pi)(s -P2) It is required that Ri and R2 be chosen so that Z3 and Z4 are RC imped- ances. Assume that each of the impedances Z3 and Zt is to be realizable as a resistor in series with a parallel combination of a resistor and capaci- tor as sliown in Fig. 9. This structure is sufficient to realize the most general first-degree RC impedance function. The ratio of these two func- tions is given by Zi (s 4- asJCs + ai) where, from (21) and (22), ''-wr^y <^) Fig. 9 — Structural form of Z3 or Zt . • It must be remembered that this model will ordinarily be inadequate for stability analyses. For this purpose it must be modified to be valid in the fre- quency range where the significant active and passive parasitic parameters are influential. SYNTHESIS OF DKIVING-POINT IMPEDANCES 957 Four possibilities exist for the pole-zero pattern of 23/Z4 as shown in Fig. 10. The zeros in Fig. 10(a) and the poles in Fig. 10(b) may occur with multiphcity two. Assume tentatively that Z(s) has complex conjugate poles and zeros. The function Z(a)\- « ^ a ^ ^], where s = a -\- jui, is nonnegative and approaches unity at both extremes of the argument. Since the func- tion is the ratio of two second-degree polyi^omials ui a, only two points J*^ (a) — ^ — e — e — ^ (b) — e — H — ^ — e- (c) — e — X — e — K- (d) — X — e — X — e- -tr Fig. 10 — Permissible pole-zero patterns for Zz/Zt . of intersection with a horizontal hne are possible. When such an inter- section occurs, the intersecting points will be separated by an cxtrcmum of the function. Hence, if Z{s') exhibits at least one extremum on the negative-real axis, the parameters Ri and R2 can be chosen to provide a pole-zero pattern for Z3/Z4 of the type shown in Figs. 10(a) or (b). Since, for both type (a) and type (b), (1 - Ri) and (1 - Sa) have the same sign, the impedances Z3 and Z4 would be realizable. Fig. 11 — Construction of the extremum points of Zia). Consider the pole-zero diagram for Z(s). It can easily be shown that the circle passing through the poles and zeros [Fig. 11(a)] will intersect the real axis at the two pomts w^here Z((j) has an extremum. Conse- quently, any open-circuit or short-circuit stable biquadratic impedance function with complex conjugate poles and zeros can be synthesized. Two permissible limithig cases exist when the zeros or poles or both the zeros and poles occur with multiplicity two on the real axis. Another 958 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1 9(l0 permissible case occurs when the zeros are at infinity and the complex conjugate poles are in the left-half plane. Suppose that Z(s) has distinct real zeros and complex conjugate poles. It can easily be shown that the circle centered on the real axis at '^o = ^T 7^> \ Pi-2 = '^p ^J^p (25) and passing through the poles [Fig. 11(b)] will intersect the real axis at the two points where Z(a) has an extremum. The circle must pass be- tween the zeros. It follows from (25) that the center of the circle will lie to the left of the zeros if the poles lie to the left of the point midway between the zeros. This will result m an extremum of Z(o-) located to the left of the poles in a region where the function is positive. It follows directly that the synthesis can be accomplished for any biquadratic impedance function with left-half plane complex conjugate poles and real zeros, where the poles are located to the left of the point midway between the zeros.* The steps in the biquadratic synthesis procedure are: (a) choose Ri and Rz so that Zs and Z4 are reahzable, and (b) from (21), identify Z3 and Zi . The permissible values of Ri and R2 can be determined by inspection of Z{a). It should be noted that this procedure is not limited to the two broad classes of functions considered above. The synthesis can obviously be accomplished if step (a) can be carried out. Hence the applicability of the procedure can be determined by inspection of Z((t). 4.1 An Example Let Z(s)= '\-J'^^ . (26) It is evident from the graph of Z(a) for this case (Fig. 12) that the choice Ri = A, R2 = G is acceptable. From (21) and (26), Z3 QZ ~ 4 9 s^ -\-is-\-i (27) Z4 4Z - 6 10s^+ ?s+ i _ 9 (s+i)(s + 2) 10 (s + i)(s+ 1)- Two pos.sibilities exist for the pair of impedances Z3 and Z4 : * Biquadratic admittance functions with distinct negative-real polea can be realizea with a negative-impedance converter as the difference of two RC ad- mittances, For this reason, such functions are not considered in detail here. SYNTHESIS OF DRIVING-POINT IMPEDANCES 959 Z, = ^ or 9 (s + 2) 10 (s + 1) ' Z^ =^ 9 (s + 2) 10 (s + I) ' Z4 = Z4 = (s + ^) (^ + 1) (s + i) ' (28) (29) 4.2 Si/nthesis on the Basis of the Network Shown in Fig. 8 The synthesis technique presented in this section can be extended to apply to the network of Fig. 8 by comparing (8) and (18) and identify- ing 7 -^'7 Zg — Z4 , Z^ = Ri . (30) V. CONCLUSION A general method of synthesizing driving-point impedances has been presented. An impedance converter is required that can be reahzed by modifying Larky's current-Inversion negative-impedance converter. An alternate realization employs Tjinvill's voltage-inversion negative-im- pedance converter and two RC impedances. This realization leads to the result that any driving-point impedance function, expressible as a real rational fraction in the complex frequency variable, can be syn- thesized as a network containing resistors, capacitors and a single nega- 2(<r) } \ / / ^ ^ ^ V \ -6 -5 -4 -3 -2 -1 Fig. 12 — Graph of Z{a) for a biquadratic function. 960 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1960 tive-impedance converter. The technique does not require the synthesis of two-port RC or ?i-port balanced resistor networks, and leads to the direct determination of the required two -terminal RC elements. The synthesis of biquadratic impedance functions has been given special attention^ resulting in structures employing the minimum number of capacitors and a moderate number of resistors. The procedure is applicable to a wide class of functions including, m particular, all open- circuit or short-circuit stable impedances with complex conjugate poles and zeros. VI. ACKNOWLEDGMENT The author is grateful to B. McMillan and to P. M. Dollard for their constructive criticism and advice. APPENDIX Transistor Circuit Analysis An approximate model for the transistor in the range where its pa- rameters are essentially independent of frequency is shown in Fig. 13 ale (e) l^ VA » O = (c) Fig. 13 — Approximate low-frequency transistor model. The collector resistance is assumed to be infinite, an approximation whichis often reasonable — especially fordrift transistors. In terms of this model, the hybrid parameters of the impedance converter circuits of Fig. 14 are Fig. 14(a): h = rid - ai)(Zz-\-Z,) " Z,(0,ay -f- 1) + Z3 + (1 - „,)ri -''' hvi - 1, ;i22 = 0, Zz_ (1 - ai) I, Z4 ai^2 \ Zi / Zz fhi — — 0+^0 (31) 1+^ 1 4- ^3 + ^1^1 ~ «l) 1 ^* ' 1o- SYNTHESI8 OF DRIVING-POINT IMPEDANCES T, 961 T, -oZ 1 o- Ui -02 1 c- Z4 -< Ca) (b) \'o- Z3 Z4 -o2 1 o- Z3 Z4 CO (d) Fig. 14 — Transistor realizations of the converter. wliere r = r. + ^^ /3= " Fig. 14(b): hn = 1 — a 1 — a -n(l - ax)(Zz + Z,) a,&,Z,- (1 - ai)(ri + Z3+ Z^) hi2 = 1 , h.2i - , 2^3 + rid - Z3 j_ r Zs+rid - ai) "| _ d - aO [, 1 n+Z, 1 Fig. 14(c): /til = (1 - ai)n ^0, ^1.12 ^ 1, '122 = 0, 0, Z4* ■02 -02 (32) 9G2 THE BELL SYSTEM TECHNICAL JOUHNAL, JULY 1960 (33) All = 0(ia2 ^ — (1 — ai) 1 + (1 — aa) -^-- — -^ 1 + (1 - 0:2; — 7? — Fig. 14(d): J, ^ _/, _ . Z3+ (1 - a2)(Z4 + r2) ^' "'^'^ a,a,Z, - {1 - aOl^a + (1 " "2) (Z4 + r.)] ^ ' /112 ^ 1, fhz = 0, /t>;.i ^ l^^^-->0^l) l3 ,_ ,, „.^A ^^A (34) Za mai - (1 - «.) Za + (1 - a2)(r2 + Z Z4 ^£}1 ^4- REFERENCES 1. Dietzold, R. L., U. S. Patent 2,5i9,065, April 17, 1951. 2. Bangert, J. T., The Tranaistor as a Network Element, B. S.T.J. , 33, March 1945, p. 329. 3. Linvill, J. G., RC Active Filters, Proc. I. RE., 42, March 1954, p. 555. 4. Horowitz, 1. M., Synthesis of Active RC Transfer i'unctions. Research Report R-507-5G, PI]i-437, Microwave Research lustitiite, Polytechnic Inst, of Brooklyn, November 195G. 5. Yanagisawii, T., RC Active Networks Using Current- Inversion-Type Nega- tive-Impei.lance Converters, Trans. I.R.E., CT-4, September 1957, p. 140, 6. Sauciberg, I. W,, Active RC Networks, Research Report R-662-58, PIB-590, Microwave Research Institute, Polytechnic Inst, of Brooklyn, Maj' 1958, 7. Kinariwala, B. K., Synthesis of Active RC Networks, B. S.T.J. , 38, September 1959, p. 1269. 8. DeClaris, N., Synthesis of Active Networks — Driving-Poiot Functions, LR.E, Nat, Conv. Rec, 1959, p. 23. 9. Larky, A. I., Negative-Impedance Converters, Trans. I.R.E., CT-4, Septem- ber 1957, p. 124. 10. Linvill, J. G,, Tranaistor Negative-Impedance Converters, Proc. I.R.E., 41, June 1953, p. 725. 11. Bode, H. W., Network Ayiabjsis and Feedback Amplifier Design, D. Van Nostrand Co., New York, 1956, p. 67. 12. Horowitz, I. M., Optimization of Negative-Impedance Conversion Methods of Active RC Synthesis, Trans. I.R.E,, CT-6, September 1959. ...i